GO is Polynominal-Space Hard
|
|
- Horatio McCarthy
- 5 years ago
- Views:
Transcription
1 GO is Polynominal-Space Hard DAVID LICHTENSTEIN AND MICHAEL SIPSER Umversay of Cahforma, Berkeley, Cahforma ABSTRACT. It ~S shown that, given an arbitrary GO posmon on an n x n board, the problem of determining the winner is Pspace hard New techmques are exploited to overcome the dffficulues ansmg from the planar nature of board games In parucular, tt ts proved that GO ts Pspace hard by reducing a Pspace-complete set, TQBF, to a game called generahzed geography, then to a planar version of that game, and finally to GO. KEY WORDS AND PHRASES computattonal complexity, games, GO, planarlty, Pspac hardness CR CATEGORIES 3 69, 5 25, Introduction A great deal of effort has been spent in the search for optimal and computationally feasible game strategies. In some cases (e.g., Bridge-it, Nim) such strategies have been found, while in others the search has been unsuccessful. Recently it has become possible to provide compelling evidence that such strategies may not always exist. For example, Even and Tarjan [4] and Schaefer [10] have shown that determining which player has a winning strategy m certain combinatorial games is a polynomial-space-complete problem [9] (See also [l, 3].) Polynomial-space-complete (Pspace-complete) problems are generally thought to be computationally infeasible because they are, in a certain sense, the most difficult to solve of all problems in Pspace (the class of problems solvable with a polynomial amount of memory on a reasonable model of computer), including the well-known class of NPcomplete problems [6]. We show that GO, a popular Oriental game with a long history, has a similar property. That is, given an arbitrary GO position on an n x n board, the problem of determining the winner is Pspace hard. (Pspace-hard problems are at least as difficult as any problem in Pspace.) To our knowledge, this is the first such result for a board game. Board games are by their nature planar--a property which frequently complicates completeness proofs. We exploit new techniques developed in [7, 8] to overcome this difficulty. In practice GO is played on a 19 x 19 board. As such it is a finite game for which a large table containing a winning strategy could, in prinople, be given. Our generalization to an n n board prevents such a solution while preserving the spirit of the game. We make no further modifications to the rules. We claim only that GO is Pspace hard rather than Pspace complete because GO is not known to be in Pspace. If there were a polynomial bound on the length of GO games, then the completeness would follow trivially. Whde it happens that actual games seem never to Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copynght notice and the Utle of the pubhcation and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery To copy otherwise, or to republish, requires a fee and/or specific permission. The work of the first author was supported by the National Soence Foundation under Grant MCS , the work of the second author was supported by the National Science Foundation under Grant MCS A01 and by an IBM Graduate Fellowship Authors' present addresses D Llchtenstem, Computer Science Division, Department of Elecmcal Engmeenng and Computer Science, Umverslty of Cahforma, Berkeley, CA 94720; M Sipser, IBM Research Laboratory, 5600 Cottle Road, San Jose, CA ACM /80/ $00 75 Journal of the Association for Computing Machinery, Vol 27, No 2, Apnl 1980, pp
2 394 D. LICHTENSTEIN AND M. SIPSER approach 19 2 moves, we are unable to argue this in general. Finally, we acknowledge that our result has no a priori relevance to the problem of determining an optimal strategy when play begins on an empty board. 2. Quantified Boolean Formulas We will prove that GO is Pspace hard by reducing a Pspace-complete set, TQBF, to the problem of deciding the winner in an arbitrary GO position. The reduction proceeds by a series of steps, first reducing TQBF to a game called generalized geography (Section 3), then to a planar version of that game (Section 4), and fmally to GO (Section 6) Definition 1. The set of quantified Boolean formulas QBF ffi {Q~v~Q2v2... Qnvn. F(vl, v2... vn) l Q, ~ {v, 3}, where the v, are Boolean variables and Fis a Boolean formula in conjunctive normal form}. Definition 2. TQBF is the set of true formulas in QBF. THEOREM 1. TQBF is Pspace complete [9]. 3. Generalized Geography Definition 3. Generalized geography (GG) is a game played by two players on the nodes of a directed graph. Play begins when the first player puts a marker on a distinguished node. In subsequent turns, players alternately place a marker on any unmarked node q such that there is a directed arc from the last node played to q. The first player who cannot move loses. This is a generalization of a commonly played game in which players must name a geographical location not yet mentioned in the game and whose first letter is the same as the last letter of the last place named. The first player to be stumped loses. This instance of geography would be modeled by a graph with as many nodes as there are places. Directed arcs would go from a node u to all those nodes whose first letters are the same as u's last letter. THEOREM 2. GG Is Pspace complete [10]. PROOF. We are given a formula B ~ QBF, B = QlvlQ2v2... QnvnF(vl, v2... vn), where we assume that Q~ = 3, Qn = V, and that Q, # Q,+~, for 1 _< i < n. we construct the graph GG(B): vi,! (,.) vi,2 Each variable v, is represented by a diamond structure, and each clause ca is represented by a single node. In addition, we have arcs (v,.2, v,+l,0) for 1 _< i < n, arcs (v,,2, ca) for 1 _< j _< m, and paths of length 2 going from ca to v,.~ for v~ in ca, and from ca to F,,~ for g in cj. V~.o is the distinguished node. Example. (3a)(Vb)(3c)(Vd)[(a +!~ + c)(b + d)] is shown in Figure I. Play proceeds rather simply. One player chooses which path to take through V-diamonds (i.e., diamonds
3 GO is Polynominal-Space Hard 395 FIGURE 1 representing universally quantified variables), and the other player chooses which path to take through 3-diamonds. After all diamonds have been traversed, the V-player chooses a clause, and the 3-player then chooses a variable from that clause. 3 then wins immediately if the chosen variable satisfies the clause; otherwise, V wins on the next move. It follows easily that 3 wins if;' B is true, and we leave the details to the reader. [] 4. Planar Generalized Geography THEOREM 3. graphs. Generahzed geography is Pspace complete, even when played on planar PROOF. (This proof is due to T. Schaefer and an anonymous referee who simplified the original construction appearing in [8] ) Every GG graph constructed in the previous section can be transformed into an equivalent planar GG graph as follows: Draw the graph in the plane, allowing arcs to cross. Pick a point in the graph where two arcs cross:.
4 396 D. LICHTENSTEIN AND M. SIPSER Replace that section of the graph by the following subgraph. o 0 [] PROPOSITION h 3 has a win in the new graph with the indicated replacement iff3 has a win m the original graph. This proposition can be proved with a simple case analysis, which we omit. To make the proof of Theorem 3 rigorous, the method of drawing a GG graph in the plane should be specified, and we refer the reader to [7, 8] for an analogous proof. COROLLARY. GG ts Pspace complete even when played on planar bipartite graphs with maximum degree 3. PROOF. The proof of this can~accomplished by making trivial modifications to the structures described above. The diamonds representing existential variables are enlarged to become: Vi, 2 Vl, 2 I VI,3 (-) Vi, 4 The back arcs representing clauses are then attached at v,,2 or ~,,2 (instead of v,,1 or ~,1), allowing the graph to remain bipartite. Nodes of indegree greater than 2 are replaced by chains, as below: / / 7 / and fan-out is handled m a similar manner. []
5 GO is Polynominal-Space Hard 397 FIG 2 An uncapturable configuration 5. The Rules of GO GO is played on a board which is a grid of 19 x 19 locations cauedpomts. There are two players, Black and White, for whom the rules are symmetric except that Black moves first. A player moves by placing a stone of his own color on a vacant point. The moves alternate between players, except that any player may pass at any time. The game terminates when both players pass. As the game progresses, the stones form clusters called groups. A group is a maximal, uniformly colored set of stones which occupy a connected region of the board A group of stones becomes surrounded if none of them is adjacent to a vacant point. After each black (white) move, all surrounded white (black) groups are removed, followed by all surrounded black (white) groups. SCORING. At the end of the game all dead stones are removed from the board. A stone is dead if it ultimately can be surrounded, despite any attempts to save it. A vacant point is said to be white territory if it is surrounded on all sides by either white stones or the edge of the board. Black territory is similarly designated. The final score for White is the count of all the white territory minus the number of white stones which have been captured (removed at any time). The black score is slmdarly calculated and the highest scorer wins. These rules are a subset of the actual rules of GO, though they are adequate for our purposes. The major omission concerns the situauon of KO which has a special rule designed to prevent infinite repetitions of the same position. A complete, concise treatment of all the rules is given in [2]. EYES. An important consequence of the GO rules is that certain configurations of stones cannot be captured. If a configuration surrounds two separated, vacant points, it is said to have two eyes. It then cannot be surrounded because it is impossible for the opponent to fill both eyes simultaneously (see Figure 2). Frequently, in the course of actual games, a player may have a nearly surrounded group of stones which he is desperately trying to connect to a group having two eyes. At the same time his opponent is trying to cut him off. We exploit such a situation later m our proof. 6. Construction of the GO Position We now encode the constructed planar generalized geography game as a GO position. We refer to the GO players as Black and White and the geography players as the 3-player and the V-player. The GO position to be constructed will have the property that Black has a winning strategy iff the 3-player has a winning strategy. The overall plan behind the construction is to have a large region of guaranteed whtte territory, together with an even larger white group of stones which is nearly surrounded (see F~gure 3). It is so large that the outcome of the game hinges on its survival; that is, Black will wm lff he can capture it. Whlte's only hope is first to escape through the small breach in the surrounding black stones and then ultimately connect to a group with two eyes This breach, however, leads to a structure which is patterned after the given geography graph. White and Black are then, in effect, forced to play the geography game with each other. (The rows of black stones extending from the left into the white group ensure that White would have no hope of recapturing any of that territory.) Each arc and vertex m the geography graph is represented by a corresponding pipe and junctton in the GO position (see Figure 4). There are essenually five types of vertices which anse in our geography graphs (see Figure 5). We gwe the corresponding GO junctions for these vertices. Note that in the generalized geography graphs which we construct, the position of a choice vertex 0.e., a vertex with outdegree >1) determines which player
6 398 D. LICHTENSTEIN AND M. SIPSER \ / \t I V guaranteed while territory FIG 3 The global picture FIG 4 Ptpes (a) (b) (c) (d) FIG. 5 (e) (a) V-player choice, (b) 3-player choice, (c) join, (d) test, (e) trivial makes the choice. This necessitates the occasional use of trivial vertices to switch the initiative. In the GO construction the nature of the junction determines which player makes the choice. Thus the trivial vertices become unnecessary and are treated as arcs. The test vertices are distinguished from the join vertices in that they occur only along the sides of the subgraphs representing variables and in the crossover boxes (see Figure 6).
7 GO is Polynominal-Space Hard (a) (b) (c) (d) F1G 6 (a) V-player choice, (b) 3-player choice, (c) joint, (d) test. The desired GO position is obtained by joining the appropriate pipes and junctions in a way which embeds the geography graph. The pipe entering the first choice junction (the first diamond) is connected to the breach m Black's wall around the large white group. White moves first.
8 400 D. LICHTENSTEIN AND M. SIPSER We now argue that if the players play "correctly," then the ensuing game will mimic a geography game in that the course of play will travel through a sequence of pipes corresponding to a valid sequence of geography arcs. Furthermore, if any player does not play correctly, his opponent will be able to win within a few moves. Upon entering or leaving any junction it will be White's turn. Inductively, we assume that the large white group is completely surrounded except for the tip of the pipe entering the current juncuon. Let us consider the case where the play is about to enter an (a) junction, corresponding to a choice by the V-player. We assume wlog that he wishes to go left. PROPOSITION 2. win in two moves. If White" s first move is not at either point 1 or point 2, then Black can PROOF. Assume that White does not play at either 1 or 2. Further assume that White does not play at 3. In that case Black plays at 2, forcing White to respond at 1, whereby Black wins at 3. If White had played at 3 initially, then by symmetry Black again has a win. [] PROPOSITION 3. If Black does not respond at point 2, then White can win in two moves. PROOF. Suppose White played at 1 and Black failed to play at 2. Then White plays at 2, capturing three black stones. Black cannot now prevent White from connecting to the White group with two eyes, and thus White wins. [] PROPOSITION 4. PROOF. Clear. [] PROPOSITION 5. White must now continue at point 3 or else lose immediately. Black must respond at point 4 or lose immediately. PROOF. Clear, using the white group which has two eyes and which is directly above 4. [] Thus if White chooses the left-hand pipe and both players play correctly, the sequence of moves would be: White--l; Black--2; White--3; Black--4. The play now continues as before, down the left-hand pipe. The large white group is again completely surrounded, except for the tip of the left-hand pipe, and it is Whlte's turn to move, fulfilling the induction assumptions. Both the (b) and the (c) junctions can be argued similarly. The (d) junction, corresponding to a selection of a variable to test by Black, is somewhat different and we analyze it here. We show that if the play enters through the right-hand pipe, then Black wins lff the play had previously passed down through the verucal pipe. PROPOSITION 6. lf playjfirst enters thts junctton at the top, then it will leave at the bottom and there will be a whzte stone placed at point 1 and a black stone at point 2. PROOF. Clear, using the white group with two eyes to force Black's move. [] PROPOSITION 7. lf play subsequently enters through the right-hand ptpe, then Black wms. PROOF. White is forced to move at 3, followed by the winning black move at 4. [] PROPOSITION 8. lf play enters through the right-hand pipe prior to entering through the top pipe, then White wins. PROOF. White moves at 3 and then has a win at either 2 or 4. [] 7. Conclusion Both the main result of this paper and an analogous result for checkers [5] are proved using planar generalized geography, and this, m turn, was initially proved using planar quantified formulas [7, 8]. It would be interesting to know if similar techniques could be
9 GO is Polynominal-Space Hard 401 used to obtain Pspace hardness proofs for games like Othello, Hex, and chess (given some "natural" n n generalization of the last). It would also be interesting to show that GO or chess is in Pspace or that either is complete for exponential time. REFERENCES! BERLEg~MP, E R, CONWAY, J, AND GUY, R Wmmng Ways Academic Press, New York. To appear 2 BLOCK, H C Axioms for GO SIGART Newsletter (ACM), 51 (April 1975), p 13 3 CHANDRA, A K, AND STOC~EYER, L J Alternation 17th Annual IEEE Syrup Found Comptr Scl, Houston, Texas, 1973, pp EVEN, S, AND TARJAN, R E A combmatonal problem which is complete In polynomial space J ACM 23, 4 (Oct 1976), FRAENKEL, A S, GAREY, M R, JOHNSON, D.S, SCHAEFER, T J, AND YESHA, Y. The complexity of checkers on an n n board 19th Annual 1EEE Symp Found Comptr ScI, Ann Arbor, Mtch, 1978, pp KARP, P M Reductblhty among combmatorlal problems In Complemty of Computer Computations, R E Mdler and J W Thatcher, Eds, Plenum Press, New York, LICHTENSTEIN, O Planar formulae and their uses. To appear m SIAM J Comptg 8 LICHTENSTEIN, D, AND SIPSER, M GO IS Pspace Hard. Memo No UCB/ERL M78/16, Electronics Res Lab., U of Cahfornia, Berkeley, Cahf MEYER, A R, AND STOCKMEYER, L J Word problems requmng exponential time Preliminary Report Proc 5th Annual ACM Symp Theory Comptg, Austin, Texas, 1973, pp 1-9 l0 SCHAEFER, T J On the complexity of some two-person perfect-information games. Comptr Syst Scz 16, 2 (ApDI 1978), RECEIVED MAY 1978; REVISED MAY 1979, ACCEPTED JUNE 1979 Journal of the Association for Computing Machinery, Vol 2% No 2 April 1980
arxiv:cs/ v2 [cs.cc] 27 Jul 2001
Phutball Endgames are Hard Erik D. Demaine Martin L. Demaine David Eppstein arxiv:cs/0008025v2 [cs.cc] 27 Jul 2001 Abstract We show that, in John Conway s board game Phutball (or Philosopher s Football),
More informationarxiv: v1 [cs.cc] 12 Dec 2017
Computational Properties of Slime Trail arxiv:1712.04496v1 [cs.cc] 12 Dec 2017 Matthew Ferland and Kyle Burke July 9, 2018 Abstract We investigate the combinatorial game Slime Trail. This game is played
More informationGeneralized Amazons is PSPACE Complete
Generalized Amazons is PSPACE Complete Timothy Furtak 1, Masashi Kiyomi 2, Takeaki Uno 3, Michael Buro 4 1,4 Department of Computing Science, University of Alberta, Edmonton, Canada. email: { 1 furtak,
More informationScrabble is PSPACE-Complete
Scrabble is PSPACE-Complete Michael Lampis 1, Valia Mitsou 2, and Karolina So ltys 3 1 KTH Royal Institute of Technology, mlampis@kth.se 2 Graduate Center, City University of New York, vmitsou@gc.cuny.edu
More informationGame Values and Computational Complexity: An Analysis via Black-White Combinatorial Games
Game Values and Computational Complexity: An Analysis via Black-White Combinatorial Games Stephen A. Fenner University of South Carolina Daniel Grier MIT Thomas Thierauf Aalen University Jochen Messner
More informationLecture 19 November 6, 2014
6.890: Algorithmic Lower Bounds: Fun With Hardness Proofs Fall 2014 Prof. Erik Demaine Lecture 19 November 6, 2014 Scribes: Jeffrey Shen, Kevin Wu 1 Overview Today, we ll cover a few more 2 player games
More informationOn the fairness and complexity of generalized k-in-a-row games
Theoretical Computer Science 385 (2007) 88 100 www.elsevier.com/locate/tcs On the fairness and complexity of generalized k-in-a-row games Ming Yu Hsieh, Shi-Chun Tsai 1001 University Road, Department of
More informationGEOGRAPHY PLAYED ON AN N-CYCLE TIMES A 4-CYCLE
GEOGRAPHY PLAYED ON AN N-CYCLE TIMES A 4-CYCLE M. S. Hogan 1 Department of Mathematics and Computer Science, University of Prince Edward Island, Charlottetown, PE C1A 4P3, Canada D. G. Horrocks 2 Department
More informationdepth parallel time width hardware number of gates computational work sequential time Theorem: For all, CRAM AC AC ThC NC L NL sac AC ThC NC sac
CMPSCI 601: Recall: Circuit Complexity Lecture 25 depth parallel time width hardware number of gates computational work sequential time Theorem: For all, CRAM AC AC ThC NC L NL sac AC ThC NC sac NC AC
More informationUNO is hard, even for a single player
UNO is hard, even for a single player The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Demaine, Erik
More informationAmazons, Konane, and Cross Purposes are PSPACE-complete
Games of No Chance 3 MSRI Publications Volume 56, 2009 Amazons, Konane, and Cross Purposes are PSPACE-complete ROBERT A. HEARN ABSTRACT. Amazons is a board game which combines elements of Chess and Go.
More informationarxiv: v2 [cs.cc] 18 Mar 2013
Deciding the Winner of an Arbitrary Finite Poset Game is PSPACE-Complete Daniel Grier arxiv:1209.1750v2 [cs.cc] 18 Mar 2013 University of South Carolina grierd@email.sc.edu Abstract. A poset game is a
More informationProblem Set 4 Due: Wednesday, November 12th, 2014
6.890: Algorithmic Lower Bounds Prof. Erik Demaine Fall 2014 Problem Set 4 Due: Wednesday, November 12th, 2014 Problem 1. Given a graph G = (V, E), a connected dominating set D V is a set of vertices such
More informationAlessandro Cincotti School of Information Science, Japan Advanced Institute of Science and Technology, Japan
#G03 INTEGERS 9 (2009),621-627 ON THE COMPLEXITY OF N-PLAYER HACKENBUSH Alessandro Cincotti School of Information Science, Japan Advanced Institute of Science and Technology, Japan cincotti@jaist.ac.jp
More informationFraser Stewart Department of Mathematics and Statistics, Xi An Jiaotong University, Xi An, Shaanxi, China
#G3 INTEGES 13 (2013) PIATES AND TEASUE Fraser Stewart Department of Mathematics and Statistics, Xi An Jiaotong University, Xi An, Shaani, China fraseridstewart@gmail.com eceived: 8/14/12, Accepted: 3/23/13,
More informationAdvanced Automata Theory 4 Games
Advanced Automata Theory 4 Games Frank Stephan Department of Computer Science Department of Mathematics National University of Singapore fstephan@comp.nus.edu.sg Advanced Automata Theory 4 Games p. 1 Repetition
More informationNim is Easy, Chess is Hard But Why??
Nim is Easy, Chess is Hard But Why?? Aviezri S. Fraenkel January 7, 2007 Department of Computer Science and Applied Mathematics Weizmann Institute of Science Rehovot 76100, Israel Abstract The game of
More informationLecture 20 November 13, 2014
6.890: Algorithmic Lower Bounds: Fun With Hardness Proofs Fall 2014 Prof. Erik Demaine Lecture 20 November 13, 2014 Scribes: Chennah Heroor 1 Overview This lecture completes our lectures on game characterization.
More information2048 IS (PSPACE) HARD, BUT SOMETIMES EASY
2048 IS (PSPE) HRD, UT SOMETIMES ESY Rahul Mehta Princeton University rahulmehta@princeton.edu ugust 28, 2014 bstract arxiv:1408.6315v1 [cs.] 27 ug 2014 We prove that a variant of 2048, a popular online
More informationSTAJSIC, DAVORIN, M.A. Combinatorial Game Theory (2010) Directed by Dr. Clifford Smyth. pp.40
STAJSIC, DAVORIN, M.A. Combinatorial Game Theory (2010) Directed by Dr. Clifford Smyth. pp.40 Given a combinatorial game, can we determine if there exists a strategy for a player to win the game, and can
More informationCrossing Game Strategies
Crossing Game Strategies Chloe Avery, Xiaoyu Qiao, Talon Stark, Jerry Luo March 5, 2015 1 Strategies for Specific Knots The following are a couple of crossing game boards for which we have found which
More informationVARIATIONS ON NARROW DOTS-AND-BOXES AND DOTS-AND-TRIANGLES
#G2 INTEGERS 17 (2017) VARIATIONS ON NARROW DOTS-AND-BOXES AND DOTS-AND-TRIANGLES Adam Jobson Department of Mathematics, University of Louisville, Louisville, Kentucky asjobs01@louisville.edu Levi Sledd
More informationUNO is hard, even for a single playe. Demaine, Erik D.; Demaine, Martin L. Citation Theoretical Computer Science, 521: 5
JAIST Reposi https://dspace.j Title UNO is hard, even for a single playe Demaine, Erik D.; Demaine, Martin L. Author(s) Nicholas J. A.; Uehara, Ryuhei; Uno, Uno, Yushi Citation Theoretical Computer Science,
More informationDomination game and minimal edge cuts
Domination game and minimal edge cuts Sandi Klavžar a,b,c Douglas F. Rall d a Faculty of Mathematics and Physics, University of Ljubljana, Slovenia b Faculty of Natural Sciences and Mathematics, University
More informationarxiv: v1 [cs.cc] 16 May 2016
On the Complexity of Connection Games Édouard Bonnet edouard.bonnet@lamsade.dauphine.fr Sztaki, Hungarian Academy of Sciences arxiv:605.0475v [cs.cc] 6 May 06 Abstract Florian Jamain florian.jamain@lamsade.dauphine.fr
More informationObliged Sums of Games
Obliged Sums of Games Thomas S. Ferguson Mathematics Department, UCLA 1. Introduction. Let g be an impartial combinatorial game. In such a game, there are two players, I and II, there is an initial position,
More informationNew Toads and Frogs Results
Games of No Chance MSRI Publications Volume 9, 1996 New Toads and Frogs Results JEFF ERICKSON Abstract. We present a number of new results for the combinatorial game Toads and Frogs. We begin by presenting
More informationContents. MA 327/ECO 327 Introduction to Game Theory Fall 2017 Notes. 1 Wednesday, August Friday, August Monday, August 28 6
MA 327/ECO 327 Introduction to Game Theory Fall 2017 Notes Contents 1 Wednesday, August 23 4 2 Friday, August 25 5 3 Monday, August 28 6 4 Wednesday, August 30 8 5 Friday, September 1 9 6 Wednesday, September
More informationarxiv: v1 [math.co] 24 Oct 2018
arxiv:1810.10577v1 [math.co] 24 Oct 2018 Cops and Robbers on Toroidal Chess Graphs Allyson Hahn North Central College amhahn@noctrl.edu Abstract Neil R. Nicholson North Central College nrnicholson@noctrl.edu
More informationarxiv: v1 [math.co] 30 Jul 2015
Variations on Narrow Dots-and-Boxes and Dots-and-Triangles arxiv:1507.08707v1 [math.co] 30 Jul 2015 Adam Jobson Department of Mathematics University of Louisville Louisville, KY 40292 USA asjobs01@louisville.edu
More information1 In the Beginning the Numbers
INTEGERS, GAME TREES AND SOME UNKNOWNS Samee Ullah Khan Department of Computer Science and Engineering University of Texas at Arlington Arlington, TX 76019, USA sakhan@cse.uta.edu 1 In the Beginning the
More informationTic-Tac-Toe on graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 72(1) (2018), Pages 106 112 Tic-Tac-Toe on graphs Robert A. Beeler Department of Mathematics and Statistics East Tennessee State University Johnson City, TN
More informationEXPLORING TIC-TAC-TOE VARIANTS
EXPLORING TIC-TAC-TOE VARIANTS By Alec Levine A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
More informationPlaying games with algorithms: Algorithmic Combinatorial Game Theory
Surveys Games of No Chance 3 MSRI Publications Volume 56, 2009 Playing games with algorithms: Algorithmic Combinatorial Game Theory ERIK D. DEMAINE AND ROBERT A. HEARN ABSTRACT. Combinatorial games lead
More informationGame Theory and Algorithms Lecture 19: Nim & Impartial Combinatorial Games
Game Theory and Algorithms Lecture 19: Nim & Impartial Combinatorial Games May 17, 2011 Summary: We give a winning strategy for the counter-taking game called Nim; surprisingly, it involves computations
More informationPositive Triangle Game
Positive Triangle Game Two players take turns marking the edges of a complete graph, for some n with (+) or ( ) signs. The two players can choose either mark (this is known as a choice game). In this game,
More informationarxiv: v1 [cs.cc] 14 Jun 2018
Losing at Checkers is Hard Jeffrey Bosboom Spencer Congero Erik D. Demaine Martin L. Demaine Jayson Lynch arxiv:1806.05657v1 [cs.cc] 14 Jun 2018 Abstract We prove computational intractability of variants
More informationPRIMES STEP Plays Games
PRIMES STEP Plays Games arxiv:1707.07201v1 [math.co] 22 Jul 2017 Pratik Alladi Neel Bhalla Tanya Khovanova Nathan Sheffield Eddie Song William Sun Andrew The Alan Wang Naor Wiesel Kevin Zhang Kevin Zhao
More informationTutorial 1. (ii) There are finite many possible positions. (iii) The players take turns to make moves.
1 Tutorial 1 1. Combinatorial games. Recall that a game is called a combinatorial game if it satisfies the following axioms. (i) There are 2 players. (ii) There are finite many possible positions. (iii)
More informationA variation on the game SET
A variation on the game SET David Clark 1, George Fisk 2, and Nurullah Goren 3 1 Grand Valley State University 2 University of Minnesota 3 Pomona College June 25, 2015 Abstract Set is a very popular card
More informationRamsey Theory The Ramsey number R(r,s) is the smallest n for which any 2-coloring of K n contains a monochromatic red K r or a monochromatic blue K s where r,s 2. Examples R(2,2) = 2 R(3,3) = 6 R(4,4)
More informationTiling Problems. This document supersedes the earlier notes posted about the tiling problem. 1 An Undecidable Problem about Tilings of the Plane
Tiling Problems This document supersedes the earlier notes posted about the tiling problem. 1 An Undecidable Problem about Tilings of the Plane The undecidable problems we saw at the start of our unit
More informationHow hard are computer games? Graham Cormode, DIMACS
How hard are computer games? Graham Cormode, DIMACS graham@dimacs.rutgers.edu 1 Introduction Computer scientists have been playing computer games for a long time Think of a game as a sequence of Levels,
More informationPlan. Related courses. A Take-Away Game. Mathematical Games , (21-801) - Mathematical Games Look for it in Spring 11
V. Adamchik D. Sleator Great Theoretical Ideas In Computer Science Mathematical Games CS 5-25 Spring 2 Lecture Feb., 2 Carnegie Mellon University Plan Introduction to Impartial Combinatorial Games Related
More informationNarrow misère Dots-and-Boxes
Games of No Chance 4 MSRI Publications Volume 63, 05 Narrow misère Dots-and-Boxes SÉBASTIEN COLLETTE, ERIK D. DEMAINE, MARTIN L. DEMAINE AND STEFAN LANGERMAN We study misère Dots-and-Boxes, where the goal
More informationTOPOLOGY, LIMITS OF COMPLEX NUMBERS. Contents 1. Topology and limits of complex numbers 1
TOPOLOGY, LIMITS OF COMPLEX NUMBERS Contents 1. Topology and limits of complex numbers 1 1. Topology and limits of complex numbers Since we will be doing calculus on complex numbers, not only do we need
More informationDice Games and Stochastic Dynamic Programming
Dice Games and Stochastic Dynamic Programming Henk Tijms Dept. of Econometrics and Operations Research Vrije University, Amsterdam, The Netherlands Revised December 5, 2007 (to appear in the jubilee issue
More informationTROMPING GAMES: TILING WITH TROMINOES. Saúl A. Blanco 1 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY x (200x), #Axx TROMPING GAMES: TILING WITH TROMINOES Saúl A. Blanco 1 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA sabr@math.cornell.edu
More informationLower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings
ÂÓÙÖÒÐ Ó ÖÔ ÐÓÖØÑ Ò ÔÔÐØÓÒ ØØÔ»»ÛÛÛº ºÖÓÛÒºÙ»ÔÙÐØÓÒ»» vol.?, no.?, pp. 1 44 (????) Lower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings David R. Wood School of Computer Science
More informationThe tenure game. The tenure game. Winning strategies for the tenure game. Winning condition for the tenure game
The tenure game The tenure game is played by two players Alice and Bob. Initially, finitely many tokens are placed at positions that are nonzero natural numbers. Then Alice and Bob alternate in their moves
More informationarxiv:cs/ v2 [cs.cc] 22 Apr 2008
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory Erik D. Demaine Robert A. Hearn arxiv:cs/0106019v2 [cs.cc] 22 Apr 2008 Abstract Combinatorial games lead to several interesting, clean
More informationEnumeration of Two Particular Sets of Minimal Permutations
3 47 6 3 Journal of Integer Sequences, Vol. 8 (05), Article 5.0. Enumeration of Two Particular Sets of Minimal Permutations Stefano Bilotta, Elisabetta Grazzini, and Elisa Pergola Dipartimento di Matematica
More informationAlgorithms. Abstract. We describe a simple construction of a family of permutations with a certain pseudo-random
Generating Pseudo-Random Permutations and Maimum Flow Algorithms Noga Alon IBM Almaden Research Center, 650 Harry Road, San Jose, CA 9510,USA and Sackler Faculty of Eact Sciences, Tel Aviv University,
More informationSurreal Numbers and Games. February 2010
Surreal Numbers and Games February 2010 1 Last week we began looking at doing arithmetic with impartial games using their Sprague-Grundy values. Today we ll look at an alternative way to represent games
More informationarxiv:cs/ v1 [cs.cc] 11 Jun 2001
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory Erik D. Demaine arxiv:cs/0106019v1 [cs.cc] 11 Jun 2001 April 25, 2008 Abstract Combinatorial games lead to several interesting, clean
More informationEasy to Win, Hard to Master:
Easy to Win, Hard to Master: Optimal Strategies in Parity Games with Costs Joint work with Martin Zimmermann Alexander Weinert Saarland University December 13th, 216 MFV Seminar, ULB, Brussels, Belgium
More informationAnalysis of Power Assignment in Radio Networks with Two Power Levels
Analysis of Power Assignment in Radio Networks with Two Power Levels Miguel Fiandor Gutierrez & Manuel Macías Córdoba Abstract. In this paper we analyze the Power Assignment in Radio Networks with Two
More informationEasy Games and Hard Games
Easy Games and Hard Games Igor Minevich April 30, 2014 Outline 1 Lights Out Puzzle 2 NP Completeness 3 Sokoban 4 Timeline 5 Mancala Original Lights Out Puzzle There is an m n grid of lamps that can be
More informationON OPTIMAL PLAY IN THE GAME OF HEX. Garikai Campbell 1 Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA 19081, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 4 (2004), #G02 ON OPTIMAL PLAY IN THE GAME OF HEX Garikai Campbell 1 Department of Mathematics and Statistics, Swarthmore College, Swarthmore,
More informationSOLITAIRE CLOBBER AS AN OPTIMIZATION PROBLEM ON WORDS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #G04 SOLITAIRE CLOBBER AS AN OPTIMIZATION PROBLEM ON WORDS Vincent D. Blondel Department of Mathematical Engineering, Université catholique
More informationAnalysis of Don't Break the Ice
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 19 Analysis of Don't Break the Ice Amy Hung Doane University Austin Uden Doane University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj
More information18 Completeness and Compactness of First-Order Tableaux
CS 486: Applied Logic Lecture 18, March 27, 2003 18 Completeness and Compactness of First-Order Tableaux 18.1 Completeness Proving the completeness of a first-order calculus gives us Gödel s famous completeness
More informationYou Should Be Scared of German Ghost
[DOI: 10.2197/ipsjjip.23.293] Regular Paper You Should Be Scared of German Ghost Erik D. Demaine 1,a) Fermi Ma 1,b) Matthew Susskind 1,c) Erik Waingarten 1,d) Received: August 1, 2014, Accepted: January
More informationThree Pile Nim with Move Blocking. Arthur Holshouser. Harold Reiter.
Three Pile Nim with Move Blocking Arthur Holshouser 3600 Bullard St Charlotte, NC, USA Harold Reiter Department of Mathematics, University of North Carolina Charlotte, Charlotte, NC 28223, USA hbreiter@emailunccedu
More information1010 Moves A move in Go is the action of a player to place his stone on a vacant intersection of the board.
Chapter 2 Basic Concepts 1000 Basic Concepts As for the rules, what was explained in the last chapter was concise enough. You will be able to start playing a game and learn more as you experience many
More informationThe Computational Complexity of Games and Puzzles. Valia Mitsou
The Computational Complexity of Games and Puzzles Valia Mitsou Abstract The subject of my thesis is studying the algorithmic properties of one and two-player games people enjoy playing, such as chess or
More informationSequential Dynamical System Game of Life
Sequential Dynamical System Game of Life Mi Yu March 2, 2015 We have been studied sequential dynamical system for nearly 7 weeks now. We also studied the game of life. We know that in the game of life,
More informationSenior Math Circles February 10, 2010 Game Theory II
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles February 10, 2010 Game Theory II Take-Away Games Last Wednesday, you looked at take-away
More informationSTRATEGY AND COMPLEXITY OF THE GAME OF SQUARES
STRATEGY AND COMPLEXITY OF THE GAME OF SQUARES FLORIAN BREUER and JOHN MICHAEL ROBSON Abstract We introduce a game called Squares where the single player is presented with a pattern of black and white
More informationa b c d e f g h 1 a b c d e f g h C A B B A C C X X C C X X C C A B B A C Diagram 1-2 Square names
Chapter Rules and notation Diagram - shows the standard notation for Othello. The columns are labeled a through h from left to right, and the rows are labeled through from top to bottom. In this book,
More informationCopyright 2010 DigiPen Institute Of Technology and DigiPen (USA) Corporation. All rights reserved.
Copyright 2010 DigiPen Institute Of Technology and DigiPen (USA) Corporation. All rights reserved. Finding Strategies to Solve a 4x4x3 3D Domineering Game BY Jonathan Hurtado B.A. Computer Science, New
More informationGame Theory and Randomized Algorithms
Game Theory and Randomized Algorithms Guy Aridor Game theory is a set of tools that allow us to understand how decisionmakers interact with each other. It has practical applications in economics, international
More informationPeeking at partizan misère quotients
Games of No Chance 4 MSRI Publications Volume 63, 2015 Peeking at partizan misère quotients MEGHAN R. ALLEN 1. Introduction In two-player combinatorial games, the last player to move either wins (normal
More informationCombined Games. Block, Alexander Huang, Boao. icamp Summer Research Program University of California, Irvine Irvine, CA
Combined Games Block, Alexander Huang, Boao icamp Summer Research Program University of California, Irvine Irvine, CA 92697 August 17, 2013 Abstract What happens when you play Chess and Tic-Tac-Toe at
More informationarxiv:cs/ v1 [cs.gt] 7 Sep 2006
Rational Secret Sharing and Multiparty Computation: Extended Abstract Joseph Halpern Department of Computer Science Cornell University Ithaca, NY 14853 halpern@cs.cornell.edu Vanessa Teague Department
More informationAdvanced Automata Theory 5 Infinite Games
Advanced Automata Theory 5 Infinite Games Frank Stephan Department of Computer Science Department of Mathematics National University of Singapore fstephan@comp.nus.edu.sg Advanced Automata Theory 5 Infinite
More informationAnalyzing Games: Solutions
Writing Proofs Misha Lavrov Analyzing Games: olutions Western PA ARML Practice March 13, 2016 Here are some key ideas that show up in these problems. You may gain some understanding of them by reading
More informationarxiv: v1 [cs.cc] 21 Jun 2017
Solving the Rubik s Cube Optimally is NP-complete Erik D. Demaine Sarah Eisenstat Mikhail Rudoy arxiv:1706.06708v1 [cs.cc] 21 Jun 2017 Abstract In this paper, we prove that optimally solving an n n n Rubik
More informationOne-Dimensional Peg Solitaire, and Duotaire
More Games of No Chance MSRI Publications Volume 42, 2002 One-Dimensional Peg Solitaire, and Duotaire CRISTOPHER MOORE AND DAVID EPPSTEIN Abstract. We solve the problem of one-dimensional Peg Solitaire.
More informationDecomposition Search A Combinatorial Games Approach to Game Tree Search, with Applications to Solving Go Endgames
Decomposition Search Combinatorial Games pproach to Game Tree Search, with pplications to Solving Go Endgames Martin Müller University of lberta Edmonton, Canada Decomposition Search What is decomposition
More information37 Game Theory. Bebe b1 b2 b3. a Abe a a A Two-Person Zero-Sum Game
37 Game Theory Game theory is one of the most interesting topics of discrete mathematics. The principal theorem of game theory is sublime and wonderful. We will merely assume this theorem and use it to
More informationA paradox for supertask decision makers
A paradox for supertask decision makers Andrew Bacon January 25, 2010 Abstract I consider two puzzles in which an agent undergoes a sequence of decision problems. In both cases it is possible to respond
More informationCombinatorial Games. Jeffrey Kwan. October 2, 2017
Combinatorial Games Jeffrey Kwan October 2, 2017 Don t worry, it s just a game... 1 A Brief Introduction Almost all of the games that we will discuss will involve two players with a fixed set of rules
More informationarxiv: v1 [cs.cc] 7 Mar 2012
The Complexity of the Puzzles of Final Fantasy XIII-2 Nathaniel Johnston Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario N1G 2W1, Canada arxiv:1203.1633v1 [cs.cc] 7 Mar
More informationMULTINATIONAL WAR IS HARD
MULTINATIONAL WAR IS HARD JONATHAN WEED Abstract. War is a simple children s game with no apparent strategy. However, players do have the ability to influence the game s outcome by deciding how to return
More informationFigure 1. Mathematical knots.
Untangle: Knots in Combinatorial Game Theory Sandy Ganzell Department of Mathematics and Computer Science St. Mary s College of Maryland sganzell@smcm.edu Alex Meadows Department of Mathematics and Computer
More informationBRITISH GO ASSOCIATION. Tournament rules of play 31/03/2009
BRITISH GO ASSOCIATION Tournament rules of play 31/03/2009 REFERENCES AUDIENCE AND PURPOSE 2 1. THE BOARD, STONES AND GAME START 2 2. PLAY 2 3. KOMI 2 4. HANDICAP 2 5. CAPTURE 2 6. REPEATED BOARD POSITION
More informationEven 1 n Edge-Matching and Jigsaw Puzzles are Really Hard
[DOI: 0.297/ipsjjip.25.682] Regular Paper Even n Edge-Matching and Jigsaw Puzzles are Really Hard Jeffrey Bosboom,a) Erik D. Demaine,b) Martin L. Demaine,c) Adam Hesterberg,d) Pasin Manurangsi 2,e) Anak
More informationOn uniquely k-determined permutations
On uniquely k-determined permutations Sergey Avgustinovich and Sergey Kitaev 16th March 2007 Abstract Motivated by a new point of view to study occurrences of consecutive patterns in permutations, we introduce
More informationConnected Identifying Codes
Connected Identifying Codes Niloofar Fazlollahi, David Starobinski and Ari Trachtenberg Dept. of Electrical and Computer Engineering Boston University, Boston, MA 02215 Email: {nfazl,staro,trachten}@bu.edu
More informationA tournament problem
Discrete Mathematics 263 (2003) 281 288 www.elsevier.com/locate/disc Note A tournament problem M.H. Eggar Department of Mathematics and Statistics, University of Edinburgh, JCMB, KB, Mayeld Road, Edinburgh
More informationOn Variants of Nim and Chomp
The Minnesota Journal of Undergraduate Mathematics On Variants of Nim and Chomp June Ahn 1, Benjamin Chen 2, Richard Chen 3, Ezra Erives 4, Jeremy Fleming 3, Michael Gerovitch 5, Tejas Gopalakrishna 6,
More informationFaithful Representations of Graphs by Islands in the Extended Grid
Faithful Representations of Graphs by Islands in the Extended Grid Michael D. Coury Pavol Hell Jan Kratochvíl Tomáš Vyskočil Department of Applied Mathematics and Institute for Theoretical Computer Science,
More informationTile Number and Space-Efficient Knot Mosaics
Tile Number and Space-Efficient Knot Mosaics Aaron Heap and Douglas Knowles arxiv:1702.06462v1 [math.gt] 21 Feb 2017 February 22, 2017 Abstract In this paper we introduce the concept of a space-efficient
More informationFormidable Fourteen Puzzle = 6. Boxing Match Example. Part II - Sums of Games. Sums of Games. Example Contd. Mathematical Games II Sums of Games
K. Sutner D. Sleator* Great Theoretical Ideas In Computer Science Mathematical Games II Sums of Games CS 5-25 Spring 24 Lecture February 6, 24 Carnegie Mellon University + 4 2 = 6 Formidable Fourteen Puzzle
More information2. The Extensive Form of a Game
2. The Extensive Form of a Game In the extensive form, games are sequential, interactive processes which moves from one position to another in response to the wills of the players or the whims of chance.
More informationThe Hex game and its mathematical side
The Hex game and its mathematical side Antonín Procházka Laboratoire de Mathématiques de Besançon Université Franche-Comté Lycée Jules Haag, 19 mars 2013 Brief history : HEX was invented in 1942
More informationChameleon Coins arxiv: v1 [math.ho] 23 Dec 2015
Chameleon Coins arxiv:1512.07338v1 [math.ho] 23 Dec 2015 Tanya Khovanova Konstantin Knop Oleg Polubasov December 24, 2015 Abstract We discuss coin-weighing problems with a new type of coin: a chameleon.
More informationPearl Puzzles are NP-complete
Pearl Puzzles are NP-complete Erich Friedman Stetson University, DeLand, FL 32723 efriedma@stetson.edu Introduction Pearl puzzles are pencil and paper puzzles which originated in Japan [11]. Each puzzle
More informationCS 491 CAP Intro to Combinatorial Games. Jingbo Shang University of Illinois at Urbana-Champaign Nov 4, 2016
CS 491 CAP Intro to Combinatorial Games Jingbo Shang University of Illinois at Urbana-Champaign Nov 4, 2016 Outline What is combinatorial game? Example 1: Simple Game Zero-Sum Game and Minimax Algorithms
More information